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Carpal Tunnel
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Carpal Tunnel
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veteran
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I checked a few things on the web and found two accounts that disagree with each other and with my friend's interpretation. One says that it's been proven that C <> 1 and another that it's been proven that we CAN'T prove whether C = 1 or not. I'm a little busy right now and don't have the time to look into it, but I am intruiged. If I can get some time over the next few months, I'll check it out.
k
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Carpal Tunnel
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Carpal Tunnel
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That's an old book by George Gamow. He starts by telling a story of two mathematical naifs who can't count above three because they have no number for it, and then goes on to develop the theory of infinities:
--aleph-null is the number of integers/reals
--aleph-one is the number of rationals/points on a line or plane or solid or...
--aleph-two is the number of possible curves in space
--There is no larger infinity that we can conceive. C, the cardinality of the Continuum, may or may not be equal to aleph-two, or might be between them.
So at least where orders of infinity are concerned, we're no more advanced than his two naifs in the story.
The book is from the 'Forties, I think, and is a delightful read. Has the math changed since?
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journeyman
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journeyman
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Ooh, I know this one.
The relationship between aleph-1 and c is called is the Continuum Hypothesis. It's now known to be independent: you can choose it to be true or false, and either is consistent with standard mathematics.
The next infinity after aleph-0 is called aleph-1, then come aleph-2, aleph-3, ..., aleph-aleph-0, ...
Nothing is known that has size aleph-1 (or any of those others bigger than aleph-0).
The set of integers has the size aleph-0. The set of real numbers has a size called c (meaning continuum), bigger than aleph-0. The set of all subsets of the integers has a size that can be expressed as 2^aleph-0 (two to the power of aleph nought), which is also bigger than aleph-0.
So we have three infinities bigger than the smallest one: aleph-1, c, and 2^aleph-0. You can prove c = 2^aleph-0. Since aleph-1 is the next biggest, either c = aleph-1 or c > aleph-1. The Continuum Hypothesis, posed by Cantor, is that they're equal.
In 1938 Gödel proved it would be consistent if the CH was true. In 1963 Paul Cohen proved it would be consistent if the contrary was true (i.e. there were infinities strictly between aleph-0 and c).
P.S. Who needs infinities? They're mathematicians: when they get to nought bottles they keep going to -1. :-)
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Pooh-Bah
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Pooh-Bah
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As an aside, a recent article in The New Yorker reviewed a book concerning infinity and Cantor. It wasn't a very glowing review but not completely scathing either.
As for myself, infinity doesn't interest me much, but the highpoint of my month has been learning that i raised to the power of i (that is, i^i) not only has a value, but it is a real number and not a complex number. The wonderfully versatile Google can be used a calculator, and if you enter i^i and hit enter you get i^i = 0.207879576.
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Carpal Tunnel
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Carpal Tunnel
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(e to the i pi) minus one equals zero.
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Pooh-Bah
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Pooh-Bah
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Yes, that is one of the most beautiful and intriguing equations in all of mathematics.
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Carpal Tunnel
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Carpal Tunnel
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Or wait a minute. Is it (e to the i pi) plus one equals zero?
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Pooh-Bah
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